Mesh-free approximations via the error reproducing kernel method and applications to nonlinear systems developing shocks

Error estimates for the error reproducing kernel method (ERKM) are provided. The ERKM is a mesh-free functional approximation scheme [A. Shaw, D. Roy, A NURBS-based error reproducing kernel method with applications in solid mechanics, Computational Mechanics (2006), to appear (available online)], wherein a targeted function and its derivatives are first approximated via non-uniform rational B-splines (NURBS) basis function. Errors in the NURBS approximation are then reproduced via a family of non-NURBS basis functions, constructed using a polynomial reproduction condition, and added to the NURBS approximation of the function obtained in the first step. In addition to the derivation of error estimates, convergence studies are undertaken for a couple of test boundary value problems with known exact solutions. The ERKM is next applied to a one-dimensional Burgers equation where, time evolution leads to a breakdown of the continuous solution and the appearance of a shock. Many available mesh-free schemes appear to be unable to capture this shock without numerical instability. However, given that any desired order of continuity is achievable through NURBS approximations, the ERKM can even accurately approximate functions with discontinuous derivatives. Moreover, due to the variation diminishing property of NURBS, it has advantages in representing sharp changes in gradients. This paper is focused on demonstrating this ability of ERKM via some numerical examples. Comparisons of some of the results with those via the standard form of the reproducing kernel particle method (RKPM) demonstrate the relative numerical advantages and accuracy of the ERKM.     

Analyses of wrinkled and slack membranes through an error reproducing mesh-free method

A mesh-free approximation of large deformations of flexible membrane structures within the tension field theory is considered in this paper. A modification of the wrinkling theory, originally proposed by Roddeman et al. (1987) [Roddeman, D.G., Drukker, J., Oomens, C.W.J., Janssen, J.D., 1987, The wrinkling of thin membranes: Part I—theory; Part II—numerical analysis. ASME J. Appl. Mech. 54, 884–892.], is proposed to study the behaviour of an isotropic membrane under the mixed state of stress (taut, wrinkled and slack). Using the facts that the state of stress is not uniform across an element and that the deformation gradient is a spatially continuous (and possibly non-differentiable) tensor, the proposed model uses a continuously modified deformation gradient to capture the location and orientation of wrinkles more precisely. While the deformation gradient need not be everywhere-differentiable in a wrinkled membrane, it is argued that the fictive non-wrinkled (non-slack) surface may be looked upon as an everywhere-taut surface in the limit as the minor (and major) principal tensile stresses over the wrinkled (slack) portions go to zero. Accordingly, the modified deformation gradient is thought of as the limit of a sequence of everywhere-differentiable tensors. The weighted residual from the governing equations are presently solved via a mesh-free method, where the entire domain is discretized only by a set of grid points. A non-uniform-rational-B-spline (NURBS) based error reproducing kernel method (ERKM) has been used to approximate the field variable over the domain. The first step in the method is to approximate a function and its derivatives through NURBS basis functions. However, since NURBS functions neither reproduce any polynomial nor interpolate the grid points (also referred to as control or nodal points), the approximated functions result in uncontrolled errors over the domain including the grid points. Accordingly the error functions in the NURBS approximation and its derivatives are reproduced via a family of non-NURBS basis functions. The non-NURBS basis functions are constructed using a polynomial reproduction condition and added to the NURBS approximation of the function obtained in the first step. Several numerical examples on wrinkled and/or slack membranes are also provided.     

An Error Criterion in Digital Image Correlation for Unknown Deformation Fields and Its Application of Parameters Selection

Digital image correlation (DIC) is a widely used optical metrology for surface deformation measurement. In DIC, the square root of the mean square error (RMS error) and standard deviation error (SD error) are used as quantitative criteria in order to evaluate the accuracy and robustness of a DIC method\algorithm. However, RMS and SD error criteria are computed from prescribed and measured displacements, which indicates that the prescribed displacement fields must be precisely generated. Therefore, it is difficult to quantitatively evaluate the accuracy and robustness of an algorithm\method in practical DIC measurements because imposed displacements are unknown (that’s why DIC measurements are needed). Moreover, the accuracy of DIC measurements highly relies on parameters selection, especially the selections of subset size and shape function. In practice, the subset size and shape function are usually selected according to experience because there are numerous factors (e.g. the quality of speckle image, local displacement field) and uncertainties (e.g. noise level, out-of-plane motion, illumination lighting fluctuation during image capturing) that affect the parameters selection, which makes it difficult to select optimal parameters based on previous works which mainly focused on theoretical deduction in ideal condition. In this paper, an error criterion for evaluating the accuracy of practical DIC measurements with unknown displacements is proposed. Numerical experiments are used to validate the effectiveness and feasibility of the proposed criterion for accuracy evaluation. It is concluded that the square root of the sum of squared forward and backward displacements difference (SFBD) error has a significant positive linear correlation with the widely used SD error in most practical DIC measurements where the mismatch between the frequently-used first- and second-order shape functions and the actual field is usually small. Also, an application of the proposed criterion is presented by real experiments for subset size and shape function selections, which verifies that the proposed error criterion can be effectively used for DIC parameters selection.     

Inverse Estimation of Cohesive Fracture Properties of Asphalt Mixtures Using an Optimization Approach

Tensile cracking in asphalt pavements due to vehicular and thermal loads has become an experimental and numerical research focus in the asphalt materials community. Previous studies have used the discrete element method (DEM) to study asphalt concrete fracture. These studies used trial-and-error to obtain local fracture properties such that the DEM models approximate the experimental load-crack mouth opening displacement response. In the current study, we identify the cohesive fracture properties of asphalt mixtures via a nonlinear optimization method. The method encompasses a comparative investigation of displacement fields obtained using both digital image correlation (DIC) and heterogeneous DEM fracture simulations. The proposed method is applied to two standard fracture test geometries: the single-edge notched beam test, SE(B), under three-point bending, and the disk-shaped compact tension test, DC(T). For each test, the Subset Splitting DIC algorithm is used to determine the displacement field in a predefined region near the notch tip. Then, a given number of DEM simulations are performed on the same specimen. The DEM is used to simulate the fracture of asphalt concrete with a linear softening cohesive contact model, where fracture-related properties (e.g., maximum tensile force and maximum crack opening) are varied within a predefined range. The difference between DIC and DEM displacement fields for each set of fracture parameters is then computed and converted to a continuous function via multivariate Lagrange interpolation. Finally, we use a Newton-like optimization technique to minimize Lagrange multinomials, yielding a set of fracture parameters that minimizes the difference between the DEM and DIC displacement fields. The optimized set of fracture parameters from this nonlinear optimization procedure led to DEM results which are consistent with the experimental results for both SE(B) and DC(T) geometries.     

Isogeometric Analysis of the Nonlinear Deformation of Planar Flexible Beams with Snap-back

Based on the continuity of the derivatives of the Non-Uniform Rational B-Splines(NURBS) curve and the Jaumann strain measure, the present paper adopted the position coordinates of the control points as the degrees of freedom and developed a planar rotation-free Euler-Bernoulli beam element for isogeometric analysis, where the derivatives of the field variables with respect to the arc-length were expressed as the sum of the weighted sum of the position coordinates of the control points, and the NURBS basis functions were used as the weight functions. Furthermore, the concept of bending strip was used to involve the rigid connection between multiple patches. Several typical examples with geometric nonlinearities were used to demonstrate the accuracy and effectiveness of the proposed algorithm. The presented formulation fully accounts for the geometric nonlinearities and can be used to study the snap-through and snap-back phenomena of flexible beams.     

Optimization of Digital Image Correlation for High-Resolution Strain Mapping of Ceramic Composites

Digital image correlation (DIC) is assessed as a tool for measuring strains with high spatial resolution in woven-fiber ceramic matrix composites. Using results of mechanical tests on aluminum alloy specimens in various geometric configurations, guidelines are provided for selecting DIC test parameters to maximize the extent of correlation and to minimize errors in displacements and strains. The latter error is shown to be exacerbated by the presence of strain gradients. In a case study, the resulting guidelines are applied to the measurement of strain fields in a SiC/SiC composite comprising 2-D woven fiber. Sub-fiber tow resolution of strain and low strain error are achieved. The fiber weave architecture is seen to exert a significant influence over strain heterogeneity within the composite. Moreover, strain concentrations at tow crossovers lead to the formation of macroscopic cracks in adjacent longitudinal tows. Such cracks initially grow stably, subject to increasing app lied stress, but ultimately lead to composite rupture. Once cracking is evident, the composite response is couched in terms of displacements, since the computed strains lack physical meaning in the vicinity of cracks. DIC is used to identify the locations of these cracks (via displacement discontinuities) and to measure the crack opening displacement profiles as a function of applied stress.     

Spatial DIC Errors due to Pattern-Induced Bias and Grey Level Discretization

Digital image correlation (DIC) is an optical metrology method widely used in experimental mechanics for full-field shape, displacement and strain measurements. The required strain resolution for engineering applications of interest mandates DIC to have a high image displacement matching accuracy, on the order of 1/100th of a pixel, which necessitates an understanding of DIC errors. In this paper, we examine two spatial bias terms that have been almost completely overlooked. They cause a persistent offset in the matching of image intensities and thus corrupt DIC results. We name them pattern-induced bias (PIB), and intensity discretization bias (IDB). We show that the PIB error occurs in the presence of an undermatched shape function and is primarily dictated by the underlying intensity pattern for a fixed displacement field and DIC settings. The IDB error is due to the quantization of the gray level intensity values in the digital camera. In this paper we demonstrate these errors and quantify their magnitudes both experimentally and with synthetic images.

     

Absolute Nodal Coordinates in Digital Image Correlation

A great deal of progress has been made in recent years in the field of global digital image correlation (DIC), where higher-order, element-based approaches were proposed to improve the interpolation performance and to better capture the displacement fields. In this research, another higher-order, element-based DIC procedure is introduced. Instead of the displacements, the elements’ global nodal positions and nodal position-vector gradients, defined according to the absolute nodal coordinate formulation, are used as the searched parameters of the Newton–Raphson iterative procedure. For the finite elements, the planar isoparametric plates with 24 nodal degrees of freedom are employed to ensure the gradients’ continuity among the elements. As such, the presented procedure imposes no linearization on the strain measure, and therefore indicates a natural consistency with the nonlinear continuum theory. To verify the new procedure and to show its advantages, a real large deformation experiment and several numerical tests on the computer-generated images are studied for the standard, low-order, element-based digital image correlation and the presented procedure. The results show that the proposed procedure proves to be accurate and reliable for describing the rigid-body movement and simple deformations, as well as for determining the continuous finite strain field of a real specimen.     

A Novel “Subset Splitting” Procedure for Digital Image Correlation on Discontinuous Displacement Fields

Digital Image Correlation (DIC) is an easy to use yet powerful approach to measure displacement and strain fields. While the method is robust and accurate for a variety of applications, standard DIC returns large error and poor correlation quality near displacement discontinuities such as cracks or shear bands. This occurs because the subsets used for correlation can only capture continuous deformations from the reference to the deformed image. As a result the regions around discontinuities are typically removed from the area of interest, before or after analysis. Here, a novel approach is proposed which enables the subset to split in two sections when a discontinuity is detected. This method enables the measurement of “displacement jumps”, and also of displacements and strains right by the discontinuity (for example a crack profile or residual strains in the wake). The method is validated on digitally created images based on mode I and mode II asymptotic displacement fields, for both sub-pixel and super-pixel crack opening displacements. Finally, an actual fracture experiment on a high density polyethylene (HDPE) specimen demonstrates the robustness of the method on actual images. Compared to other methods capable of handling discontinuities, this novel “subset-splitting” procedure offers the advantage of being a direct extension of the now popular standard DIC, and can therefore be implemented as an “upgrade” to that method.     

Use of 3-D Digital Image Correlation to Characterize the Mechanical Behavior of a Fiber Reinforced Refractory Castable

Refractory castables exhibit very low fracture strain levels when subjected to tension or bending. The main objective of this work is to show that 3-D digital image correlation (3-D DIC) allows such low strain levels to be measured. Compared to mechanical extensometer measurements, 3-D DIC makes it possible to reach similar strain resolution levels and to avoid the problem of position dependance related to the heterogeneous nature of the strain and to strain localization phenomena. First, the 3-D DIC method and the experimental set-up are presented. Secondly, an analysis of the 3-D DIC method is performed in order to evaluate the resolution, the standard uncertainty and the spatial resolution for both displacement and strain measurements. An optimized compromise between strain spatial resolution and standard uncertainty is reached for the configuration of the experimental bending test. Finally, the macroscopic mechanical behavior of a fiber reinforced refractory castable (FRRC) is studied using mechanical extensometry and 3-D DIC in the case of tensile and four-point bending tests. It is shown that similar results are obtained with both methods. Furthermore, in the case of bending tests on damaged castable, 3-D DIC results demonstrate the ability to determine Young’s modulus from heterogeneous strain fields better than by using classical beam deflection measurements.     

数字图像相关方法中匹配及过匹配形函数的误差分析

基于图像子区的数字图像相关方法需采用合适的形函数来近似目标图像子区的真实变形。由于实际测量时目标子区的局部变形往往是未知的,实际采用的不同阶次(零阶、一阶和二阶)的形函数不可避免地产生误匹配(欠匹配和过匹配)问题,从而引入位移测量的系统或随机误差。尽管由欠匹配形函数引起的系统误差已被充分认识,由过匹配形函数引起的位移误差仍缺少理论解释。本文首先推导出采用一阶和二阶形函数的数字图像相关方法的随机误差理论公式,随后采用一系列数值实验验证了理论公式的准确性。结果显示:过匹配形函数不会引入额外的系统误差,但会增加随机误差,且二阶形函数的随机误差是一阶形函数的二倍。考虑到由欠匹配一阶形函数引入的系统误差往往远大于过匹配二阶形函数的随机误差,因此在未能确知变形的情况下,推荐使用二阶形函数。     

Isogeometric analysis of free-form Timoshenko curved beams including the nonlinear effects of large deformations

In the present paper, the isogeometric analysis (IGA) of free-form planar curved beams is formulated based on the nonlinear Timoshenko beam theory to investigate the large deformation of beams with variable curvature. Based on the isoparametric concept, the shape functions of the field variables (displacement and rotation) in a finite element analysis are considered to be the same as the non-uniform rational basis spline (NURBS) basis functions defining the geometry. The validity of the presented formulation is tested in five case studies covering a wide range of engineering curved structures including from straight and constant curvature to variable curvature beams. The nonlinear deformation results obtained by the presented method are compared to well-established benchmark examples and also compared to the results of linear and nonlinear finite element analyses. As the nonlinear load-deflection behavior of Timoshenko beams is the main topic of this article, the results strongly show the applicability of the IGA method to the large deformation analysis of free-form curved beams. Finally, it is interesting to notice that, until very recently, the large deformations analysis of free-form Timoshenko curved beams has not been considered in IGA by researchers.     

材料力学性能显微测试系统

介绍了适用于毫、微米量级材料的力学性能测试系统,该系统包括微型加力装置、显微镜及数字图像处理系统。通过计算机控制,加力机构可以对微小试件按力或位移控制方式加载,载荷大小通过传感器直接由计算机数字化。试件的变形测量采用数字图像相关技术,通过序列图像分析方式,获得试件在不同外力作用下对应的变形,从而获得该材料的应力应变关系。在利用显微镜对微细观图像进行放大时,图像的畸变会影响相关计算的精度,本文采用了精度极高的正交栅板,对成像系统进行标定,有效地消除了高放大倍数下的图像畸变。图像相关技术对被测物体的表面处理非常敏感,本文也对这一问题进行了讨论,并给出两个不同的应用实例。第一个为材料性能试验,对厚度为0.1mm的铝材进行单向拉伸,给出了材料的应力应变关系;第二个为稳态裂纹扩展试验,对用人牙制作的紧凑拉伸试件进行测试,获得裂尖后的COD曲线。实验表明,该测试系统在微细观力学实验中具有广阔的应用前景。     

Direct Extraction of Cohesive Fracture Properties from Digital Image Correlation: A Hybrid Inverse Technique

The accuracy of an adopted cohesive zone model (CZM) can affect the simulated fracture response significantly. The CZM has been usually obtained using global experimental response, e.g., load versus either crack opening displacement or load-line displacement. Apparently, deduction of a local material property from a global response does not provide full confidence of the adopted model. The difficulties are: (1) fundamentally, stress cannot be measured directly and the cohesive stress distribution is non-uniform; (2) accurate measurement of the full crack profile (crack opening displacement at every point) is experimentally difficult to obtain. An attractive feature of digital image correlation (DIC) is that it allows relatively accurate measurement of the whole displacement field on a flat surface. It has been utilized to measure the mode I traction-separation relation. A hybrid inverse method based on combined use of DIC and finite element method is used in this study to compute the cohesive properties of a ductile adhesive, Devcon Plastic Welder II, and a quasi-brittle plastic, G-10/FR4 Garolite. Fracture tests were conducted on single edge-notched beam specimens (SENB) under four-point bending. A full-field DIC algorithm was employed to compute the smooth and continuous displacement field, which is then used as input to a finite element model for inverse analysis through an optimization procedure. The unknown CZM is constructed using a flexible B-spline without any “a priori” assumption on the shape. The inversely computed CZMs for both materials yield consistent results. Finally, the computed CZMs are verified through fracture simulation, which shows good experimental agreement.     

基于双线性位移模式数字图像相关方法的误差分析及应用

探讨数字图像相关方法测试结果的误差分布规律对提高该方法的测试精度具有重要意义。本文从理论上分析了基于双线性位移模式数字图像相关方法的子区变形场测试误差分布规律,结果表明,当试件的真实变形可由双线性位移模式描述时,变形场最大测试误差通常出现在子区的边界或节点处。因此,若试件发生均匀拉伸等常应变变形,可利用相关系数选取一个最优子区,认为测得最优子区中心应变为试件真实应变。零变形实验验证了该测试方法的可靠性。最后对手机导光板试件在单轴拉伸载荷下的数字图像进行分析,并利用本文方法测试了其弹性常数。     

Mesh-Based Digital Image Correlation Method Using Non-Uniform Elements for Measuring Displacement Fields with High Gradient

A new mesh-based digital image correlation (mesh-DIC) scheme using non-uniform 8-node quadrilateral (Q8) elements is proposed to process displacement fields with regions of both high and low gradient. The effect of element size on Q8-DIC is extensively analyzed, and the results show that when the shape function of an element matches or over-matches the gradient of the displacement field, a larger element size can achieve better results. Conversely, when the shape function under-matches the gradient of the displacement field, a suitable element size should be chosen. Furthermore, a practical and effective method based on the residual error of the DIC calculation is proposed to automatically refine mesh with non-uniform elements. The effectiveness of this method is verified using simulated speckle images that contain one- and two-dimensional non-uniform displacements. The proposed method is also used to improve the accuracy of two experiments that measure displacement fields with high gradient.     

Mixed numerical–experimental technique for orthotropic parameter identification using biaxial tensile tests on cruciform specimens

This paper presents a mixed numerical–experimental method for the identification of the four in-plane orthotropic engineering constants of composite plate materials. A biaxial tensile test is performed on a cruciform test specimen. The heterogeneous displacement field is observed by a CCD camera and measured by a digital image correlation (DIC) technique. The measured displacement field and the subsequently computed strain field are compared with a finite element simulation of the same experiment. The four independent engineering constants are unknown parameters in the finite element model. Starting from an initial value, these parameters are updated till the computed strain field matches the experimental strain field. Two specimen geometries are used: one with a centered hole to increase the strain heterogeneity and one without a hole. It is found that the non-perforated specimen yields the most accurate results.     

Grid Method for Microscale Discontinuous Deformation Measurement

The objective of this paper is to explore both grid method and Digital Image Correlation (DIC) technique for microscale and discontinuous displacement measurements, such as those associated with crack tips. First, the principle of the grid method is revisited. The grid method and DIC technique are then applied to computer generated images to calculate the displacement field around crack tips. Finally, the grid method is applied to actual experimental images of fracture tests which are conducted inside a Scanning Electron Microscope (SEM) chamber. A new technique is developed to generate microscale pattern that is suitable for both grid method and DIC technique. The displacement fields calculated from grid method are compared with those from DIC technique to identify the strengths and weaknesses of each technique for the microscale and discontinuous displacement measurements. It has been determined that grid method can obtain data closer to the discontinuity than DIC; however, DIC produces smoother displacement fields at the far field. Using this new pattern generation technique, both grid method and DIC technique can be applied to the fracture test at the microscale to complement with each other to achieve the best experiment results.     

基于NURBS插值的三维渐进结构优化方法

提出一种基于等几何控制点密度变量的三维双向渐进结构拓扑优化方法。在当前列式下,高阶NURBS基函数被同时用于CAD模型中NURBS实体片的几何场、位移场和温度场以及密度场插值,实现了几何模型、分析模型和优化模型的有效统一,确保了位移场、温度场及密度场的高阶连续性;详细推导了基于等几何控制点密度变量的三维渐进结构法模型及其灵敏度分析列式;最后几个典型的数值算例,包括最小柔顺性、热传导优化问题及三维结构自由振动的基频最大化问题,验证了本文方法的有效性。     

Assessment of Digital Image Correlation Measurement Errors: Methodology and Results

Optical full-field measurement methods such as Digital Image Correlation (DIC) are increasingly used in the field of experimental mechanics, but they still suffer from a lack of information about their metrological performances. To assess the performance of DIC techniques and give some practical rules for users, a collaborative work has been carried out by the Workgroup “Metrology” of the French CNRS research network 2519 “MCIMS (Mesures de Champs et Identification en Mécanique des Solides / Full-field measurement and identification in solid mechanics, http://www.ifma.fr/lami/gdr2519)”. A methodology is proposed to assess the metrological performances of the image processing algorithms that constitute their main component, the knowledge of which being required for a global assessment of the whole measurement system. The study is based on displacement error assessment from synthetic speckle images. Series of synthetic reference and deformed images with random patterns have been generated, assuming a sinusoidal displacement field with various frequencies and amplitudes. Displacements are evaluated by several DIC packages based on various formulations and used in the French community. Evaluated displacements are compared with the exact imposed values and errors are statistically analyzed. Results show general trends rather independent of the implementations but strongly correlated with the assumptions of the underlying algorithms. Various error regimes are identified, for which the dependence of the uncertainty with the parameters of the algorithms, such as subset size, gray level interpolation or shape functions, is discussed.